Mechanical Vibrations

Vibration is the periodic motion of a body or system of connected bodies displaced from a position of equilibrium.

Free Undamped Vibrations

Consider a block of mass mm attached to a spring of stiffness kk.

Equation of Motion

For free vibration (no external force), the equation of motion is:
mx¨+kx=0m \ddot{x} + k x = 0
The solution is simple harmonic motion:
x(t)=Asin⁡(ωnt)+Bcos⁡(ωnt)x(t) = A \sin(\omega_n t) + B \cos(\omega_n t)

Vibration Parameters

  • Natural Circular Frequency (ωn\omega_n): The rate of oscillation in radians per second. ωn=km(rad/s)\omega_n = \sqrt{\frac{k}{m}} \quad (\text{rad/s})
  • Natural Frequency (fnf_n): The number of cycles per second. fn=ωn2π(Hz)f_n = \frac{\omega_n}{2\pi} \quad (\text{Hz})
  • Period (τ\tau): The time required for one complete cycle. τ=1fn=2πωn=2πmk(s)\tau = \frac{1}{f_n} = \frac{2\pi}{\omega_n} = 2\pi \sqrt{\frac{m}{k}} \quad (\text{s})
Interact with the simulation below to explore mechanical vibrations.

Mechanical Vibrations Simulator

Underdamped (ζ=0.06\zeta = 0.06)
Mass (mm)5 kg
Stiffness (kk)50 N/m
Damping (cc)2 Ns/m
Initial Disp (x0x_0)1 m
Natural Freq (ωn\omega_n):3.16 rad/s
Critical Damping (ccc_c):31.62 Ns/m
Damped Freq (ωd\omega_d):3.16 rad/s

Simple Pendulum

Interact with the simulation below to explore simple pendulum motion.

Simple Pendulum Simulator

Period (TT)2.46 s
Frequency (ff)0.41 Hz
Angular Freq (ωn\omega_n)2.56 rad/s
For small angles (sin⁡θ≈θ\sin \theta \approx \theta), a simple pendulum behaves like a spring-mass system.

Pendulum Formulas

  • Natural Frequency: ωn=gL\omega_n = \sqrt{\frac{g}{L}}
  • Period: τ=2πLg\tau = 2\pi \sqrt{\frac{L}{g}}

Torsional Vibrations

Vibrations can also occur in rotational systems. The angular equivalent to a spring is a torsional spring (e.g., a twisting shaft), which provides a restoring torque MtM_t proportional to the angle of twist θ\theta.

Torsional Vibration Equation

Mt=−ktθM_t = -k_t \theta
Where ktk_t is the torsional stiffness (torque per unit angle). For a disk with mass moment of inertia I0I_0 attached to a torsional spring, the equation of motion is derived from ∑MO=IOα\sum M_O = I_O \alpha:
−ktθ=I0θ¨-k_t \theta = I_0 \ddot{\theta}
Rearranging yields the standard form:
I0θ¨+ktθ=0I_0 \ddot{\theta} + k_t \theta = 0
The natural circular frequency for torsional vibration is:
ωn=ktI0\omega_n = \sqrt{\frac{k_t}{I_0}}

Damped Free Vibrations

If a viscous damper (cc) is added to the system, energy is dissipated. The equation of motion becomes:

Damped Vibration Equation

mx¨+cx˙+kx=0m \ddot{x} + c \dot{x} + k x = 0

Damping Parameters

  • Critical Damping Coefficient (ccc_c): The minimum damping required to prevent oscillation. cc=2mωn=2kmc_c = 2m\omega_n = 2\sqrt{km}
  • Damping Ratio (ζ\zeta): A dimensionless measure of damping. ζ=ccc\zeta = \frac{c}{c_c}
System Behavior:
  • Underdamped (ζ<1\zeta \lt 1): Oscillates with decreasing amplitude.
  • Critically Damped (ζ=1\zeta = 1): Returns to equilibrium as quickly as possible without oscillating.
  • Overdamped (ζ>1\zeta \gt 1): Returns to equilibrium slowly without oscillating.

Logarithmic Decrement

For an underdamped system (ζ<1\zeta \lt 1), the amplitude of oscillation decays exponentially. A practical way to measure the damping ratio experimentally is by comparing successive peak amplitudes.

Logarithmic Decrement (δ\delta)

The logarithmic decrement δ\delta is defined as the natural logarithm of the ratio of two successive peak amplitudes, x1x_1 and x2x_2 (separated by one full cycle).
δ=ln⁡(x1x2)=2πζ1−ζ2\delta = \ln\left(\frac{x_1}{x_2}\right) = \frac{2\pi \zeta}{\sqrt{1 - \zeta^2}}
For very small damping (ζ≪1\zeta \ll 1), this equation simplifies to:
δ≈2πζ\delta \approx 2\pi \zeta
If you measure peaks over nn cycles (from peak x0x_0 to peak xnx_n), the decrement can also be found as:
δ=1nln⁡(x0xn)\delta = \frac{1}{n} \ln\left(\frac{x_0}{x_n}\right)

Forced Vibrations

When a vibrating system is subjected to a continuous external periodic force, it undergoes forced vibration. Let the driving force be F(t)=F0sin⁡(ω0t)F(t) = F_0 \sin(\omega_0 t), where ω0\omega_0 is the driving frequency.

Forced Undamped Vibration

The equation of motion is:
mx¨+kx=F0sin⁡(ω0t)m \ddot{x} + k x = F_0 \sin(\omega_0 t)
The steady-state solution (after transient free vibration decays) is:
xp(t)=Xsin⁡(ω0t)x_p(t) = X \sin(\omega_0 t)
Where the amplitude XX is given by:
X=F0k−mω02=F0/k1−(ω0/ωn)2X = \frac{F_0}{k - m\omega_0^2} = \frac{F_0/k}{1 - (\omega_0/\omega_n)^2}
Magnification Factor (MF): The ratio of the dynamic amplitude XX to the static deflection xstatic=F0/kx_{static} = F_0/k.
MF=XF0/k=1∣1−(ω0/ωn)2∣MF = \frac{X}{F_0/k} = \frac{1}{|1 - (\omega_0/\omega_n)^2|}

Important

Resonance: When the driving frequency ω0\omega_0 exactly equals the natural frequency ωn\omega_n, the amplitude theoretically goes to infinity (if undamped). This condition can cause severe structural damage.
Key Takeaways
  • Free Vibration (mx¨+kx=0m\ddot{x} + kx = 0) occurs without external forces and results in simple harmonic motion.
  • Natural Frequency (ωn=k/m\omega_n = \sqrt{k/m}) depends only on the system's mass and stiffness.
  • Torsional Vibration follows the same principles, replacing mass with mass moment of inertia (I0I_0) and linear stiffness with torsional stiffness (ktk_t).
  • Period: τ=2π/ωn\tau = 2\pi/\omega_n
  • Damping (cc) dissipates energy and causes amplitude decay.
  • Logarithmic Decrement (δ\delta) is used to measure damping from the decay of successive amplitude peaks.
  • Critical Damping (ccc_c) defines the boundary between oscillatory and non-oscillatory motion.
  • Resonance occurs in forced vibration when driving frequency equals natural frequency, causing unbounded amplitude.
PreviousThree-Dimensional Kinetics of Rigid Bodies - Examples & Applications
Quiz Me
NextMechanical Vibrations - Examples & Applications